Theorem f1omo 48881

Description: There is at most one element in the function value of a constant function whose output is 1_o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 48880 assuming ax-un 7675 (see f1omoALT 48883). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)

Hypothesis

Ref	Expression
f1omo.1	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omo	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo

Step	Hyp	Ref	Expression
1		1oex 8405	. . . 4 ⊢ 1o ∈ V
2		eqid 2729	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 48879	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 48802	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		df1o2 8402	. . . . . 6 ⊢ 1o = {∅}
6	5	eqeq2i 2742	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o ↔ ((𝐴 × {1o})‘𝑋) = {∅})
7		mosn 48801	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = {∅} → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
8	6, 7	sylbi 217	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
9	4, 8	jaoi 857	. . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
10	3, 9	ax-mp 5	. 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
11		f1omo.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
12	11	fveq1d 6828	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
13	12	eleq2d 2814	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
14	13	mobidv 2542	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
15	10, 14	mpbiri 258	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  ∅c0 4286  {csn 4579   × cxp 5621  ‘cfv 6486  1_oc1o 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-1o 8395

This theorem is referenced by:  indthinc  49451  prsthinc  49453